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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 10:30, 4 May 2009
, 10:30, 4 May 2009→Example 6
====Example 6====
====Example 6====
We have now developed two ways of computing a logical involution that raises a 2-adic relative term to the power of a 1-adic absolute term, for example, <math>\mathit{l}^\mathrm{w}\!</math> for "lover of every woman".
The first method operates in the medium of set theory, expressing the denotation of the term <math>\mathit{l}^\mathrm{w}\!</math> as the intersection of a set of relational applications:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x</math>
| height="60" | <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x</math>
|}
The second method operates in the matrix representation, expressing the value of the matrix <math>\mathfrak{L}^\mathfrak{w}</math> at an argument <math>u\!</math> as a product of coefficient powers:
{| align="center" cellspacing="6" width="90%"
|-
|-
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_u ~=~ \prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math>
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_u ~=~ \prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math>
|}
|}
Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved. The Figure below represents a universe of discourse <math>X\!</math> that is subject to the following data:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"